Theorem 1cvrat 34762

Description: Create an atom under an element covered by the lattice unit. Part of proof of Lemma B in [Crawley] p. 112. (Contributed by NM, 30-Apr-2012.)

Hypotheses

Ref	Expression
1cvrat.b	|- B = ( Base ` K )
1cvrat.l	|- .<_ = ( le ` K )
1cvrat.j	|- .\/ = ( join ` K )
1cvrat.m	|- ./\ = ( meet ` K )
1cvrat.u	|- .1. = ( 1. ` K )
1cvrat.c	|- C = ( <o ` K )
1cvrat.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
1cvrat	|- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> ( ( P .\/ Q ) ./\ X ) e. A )

Proof of Theorem 1cvrat

Step	Hyp	Ref	Expression
1		hllat 34650	. . . . . 6 |- ( K e. HL -> K e. Lat )
2	1	3ad2ant1 1082	. . . . 5 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> K e. Lat )
3		simp21 1094	. . . . . 6 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> P e. A )
4		1cvrat.b	. . . . . . 7 |- B = ( Base ` K )
5		1cvrat.a	. . . . . . 7 |- A = ( Atoms ` K )
6	4, 5	atbase 34576	. . . . . 6 |- ( P e. A -> P e. B )
7	3, 6	syl 17	. . . . 5 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> P e. B )
8		simp22 1095	. . . . . 6 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> Q e. A )
9	4, 5	atbase 34576	. . . . . 6 |- ( Q e. A -> Q e. B )
10	8, 9	syl 17	. . . . 5 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> Q e. B )
11		1cvrat.j	. . . . . 6 |- .\/ = ( join ` K )
12	4, 11	latjcom 17059	. . . . 5 |- ( ( K e. Lat /\ P e. B /\ Q e. B ) -> ( P .\/ Q ) = ( Q .\/ P ) )
13	2, 7, 10, 12	syl3anc 1326	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> ( P .\/ Q ) = ( Q .\/ P ) )
14	13	oveq1d 6665	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> ( ( P .\/ Q ) ./\ X ) = ( ( Q .\/ P ) ./\ X ) )
15	4, 11	latjcl 17051	. . . . 5 |- ( ( K e. Lat /\ Q e. B /\ P e. B ) -> ( Q .\/ P ) e. B )
16	2, 10, 7, 15	syl3anc 1326	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> ( Q .\/ P ) e. B )
17		simp23 1096	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> X e. B )
18		1cvrat.m	. . . . 5 |- ./\ = ( meet ` K )
19	4, 18	latmcom 17075	. . . 4 |- ( ( K e. Lat /\ ( Q .\/ P ) e. B /\ X e. B ) -> ( ( Q .\/ P ) ./\ X ) = ( X ./\ ( Q .\/ P ) ) )
20	2, 16, 17, 19	syl3anc 1326	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> ( ( Q .\/ P ) ./\ X ) = ( X ./\ ( Q .\/ P ) ) )
21	14, 20	eqtrd 2656	. 2 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> ( ( P .\/ Q ) ./\ X ) = ( X ./\ ( Q .\/ P ) ) )
22		simp1 1061	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> K e. HL )
23	17, 8, 3	3jca 1242	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> ( X e. B /\ Q e. A /\ P e. A ) )
24		simp31 1097	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> P =/= Q )
25	24	necomd 2849	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> Q =/= P )
26		simp33 1099	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> -. P .<_ X )
27		hlop 34649	. . . . . 6 |- ( K e. HL -> K e. OP )
28	27	3ad2ant1 1082	. . . . 5 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> K e. OP )
29		1cvrat.l	. . . . . 6 |- .<_ = ( le ` K )
30		1cvrat.u	. . . . . 6 |- .1. = ( 1. ` K )
31	4, 29, 30	ople1 34478	. . . . 5 |- ( ( K e. OP /\ Q e. B ) -> Q .<_ .1. )
32	28, 10, 31	syl2anc 693	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> Q .<_ .1. )
33		simp32 1098	. . . . 5 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> X C .1. )
34		1cvrat.c	. . . . . 6 |- C = ( <o ` K )
35	4, 29, 11, 30, 34, 5	1cvrjat 34761	. . . . 5 |- ( ( ( K e. HL /\ X e. B /\ P e. A ) /\ ( X C .1. /\ -. P .<_ X ) ) -> ( X .\/ P ) = .1. )
36	22, 17, 3, 33, 26, 35	syl32anc 1334	. . . 4 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> ( X .\/ P ) = .1. )
37	32, 36	breqtrrd 4681	. . 3 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> Q .<_ ( X .\/ P ) )
38	4, 29, 11, 18, 5	cvrat3 34728	. . . 4 |- ( ( K e. HL /\ ( X e. B /\ Q e. A /\ P e. A ) ) -> ( ( Q =/= P /\ -. P .<_ X /\ Q .<_ ( X .\/ P ) ) -> ( X ./\ ( Q .\/ P ) ) e. A ) )
39	38	imp 445	. . 3 |- ( ( ( K e. HL /\ ( X e. B /\ Q e. A /\ P e. A ) ) /\ ( Q =/= P /\ -. P .<_ X /\ Q .<_ ( X .\/ P ) ) ) -> ( X ./\ ( Q .\/ P ) ) e. A )
40	22, 23, 25, 26, 37, 39	syl23anc 1333	. 2 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> ( X ./\ ( Q .\/ P ) ) e. A )
41	21, 40	eqeltrd 2701	1 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ X e. B ) /\ ( P =/= Q /\ X C .1. /\ -. P .<_ X ) ) -> ( ( P .\/ Q ) ./\ X ) e. A )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   meetcmee 16945   1.cp1 17038   Latclat 17045   OPcops 34459   <o ccvr 34549   Atomscatm 34550   HLchlt 34637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-p1 17040  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638

This theorem is referenced by:  cdlemblem  35079  cdlemb  35080  lhpat  35329